\(\int \frac {(7+3 x-6 x^2)^q (3+2 x+4 x^2)}{(1+5 x-2 x^2)^2} \, dx\) [31]

Optimal result

Integrand size = 35, antiderivative size = 448 \[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=-\frac {(23+14 x) \left (7+3 x-6 x^2\right )^{1+q}}{264 \left (1+5 x-2 x^2\right )}+\frac {3^{-\frac {3}{2}+2 q} \left (224-1665 q+327 \sqrt {33} q\right ) \left (\frac {3-\sqrt {177}-12 x}{5-\sqrt {33}-4 x}\right )^{-q} \left (\frac {3+\sqrt {177}-12 x}{5-\sqrt {33}-4 x}\right )^{-q} \left (7+3 x-6 x^2\right )^q \operatorname {AppellF1}\left (-2 q,-q,-q,1-2 q,\frac {12-3 \sqrt {33}-\sqrt {177}}{3 \left (5-\sqrt {33}-4 x\right )},\frac {12-3 \sqrt {33}+\sqrt {177}}{3 \left (5-\sqrt {33}-4 x\right )}\right )}{352 \sqrt {11} q}-\frac {3^{-\frac {3}{2}+2 q} \left (224-3 \left (555+109 \sqrt {33}\right ) q\right ) \left (\frac {3-\sqrt {177}-12 x}{5+\sqrt {33}-4 x}\right )^{-q} \left (\frac {3+\sqrt {177}-12 x}{5+\sqrt {33}-4 x}\right )^{-q} \left (7+3 x-6 x^2\right )^q \operatorname {AppellF1}\left (-2 q,-q,-q,1-2 q,\frac {12+3 \sqrt {33}-\sqrt {177}}{3 \left (5+\sqrt {33}-4 x\right )},\frac {12+3 \sqrt {33}+\sqrt {177}}{3 \left (5+\sqrt {33}-4 x\right )}\right )}{352 \sqrt {11} q}-\frac {7}{11} 2^{-4-3 q} 59^q (1+2 q) (1-4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right ) \] Output:

-1/264*(23+14*x)*(-6*x^2+3*x+7)^(1+q)/(-2*x^2+5*x+1)+1/3872*3^(-3/2+2*q)*( 
224-1665*q+327*33^(1/2)*q)*(-6*x^2+3*x+7)^q*AppellF1(-2*q,-q,-q,1-2*q,(12- 
3*33^(1/2)-177^(1/2))/(15-3*33^(1/2)-12*x),(12-3*33^(1/2)+177^(1/2))/(15-3 
*33^(1/2)-12*x))*11^(1/2)/q/(((3-177^(1/2)-12*x)/(5-33^(1/2)-4*x))^q)/(((3 
+177^(1/2)-12*x)/(5-33^(1/2)-4*x))^q)-1/3872*3^(-3/2+2*q)*(224-3*(555+109* 
33^(1/2))*q)*(-6*x^2+3*x+7)^q*AppellF1(-2*q,-q,-q,1-2*q,(12+3*33^(1/2)-177 
^(1/2))/(15+3*33^(1/2)-12*x),(12+3*33^(1/2)+177^(1/2))/(15+3*33^(1/2)-12*x 
))*11^(1/2)/q/(((3-177^(1/2)-12*x)/(5+33^(1/2)-4*x))^q)/(((3+177^(1/2)-12* 
x)/(5+33^(1/2)-4*x))^q)-7/11*2^(-4-3*q)*59^q*(1+2*q)*(1-4*x)*hypergeom([1/ 
2, -q],[3/2],3/59*(1-4*x)^2)
 

Mathematica [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx \] Input:

Integrate[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^2,x]
 

Output:

Integrate[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^2, x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (4 x^2+2 x+3\right ) \left (-6 x^2+3 x+7\right )^q}{\left (-2 x^2+5 x+1\right )^2} \, dx\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\int \frac {88 \left (-6 x^2+3 x+7\right )^q \left (-84 (2 q+1) x^2+6 (35-39 q) x+69 q+154\right )}{-2 x^2+5 x+1}dx}{23232}-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{264 \left (-2 x^2+5 x+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{264} \int \frac {\left (-6 x^2+3 x+7\right )^q \left (-84 (2 q+1) x^2+6 (35-39 q) x+69 q+154\right )}{-2 x^2+5 x+1}dx-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{264 \left (-2 x^2+5 x+1\right )}\)

\(\Big \downarrow \) 7279

\(\displaystyle \frac {1}{264} \int \left (42 (2 q+1) \left (-6 x^2+3 x+7\right )^q+\frac {(-654 x q-15 q+112) \left (-6 x^2+3 x+7\right )^q}{-2 x^2+5 x+1}\right )dx-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{264 \left (-2 x^2+5 x+1\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{264} \left (\int \frac {(-654 x q-15 q+112) \left (-6 x^2+3 x+7\right )^q}{-2 x^2+5 x+1}dx-21\ 2^{-3 q-1} 59^q (2 q+1) (1-4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right )\right )-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{264 \left (-2 x^2+5 x+1\right )}\)

Input:

Int[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2135

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( 
q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( 
(A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b 
*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* 
e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 
*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))   Int[(a + b*x + c 
*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 
 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + 
 a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p 
+ 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B 
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a 
*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p 
 + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( 
c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) 
*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d 
 - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d 
 - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) 
 &&  !IGtQ[q, 0]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {\left (-6 x^{2}+3 x +7\right )^{q} \left (4 x^{2}+2 x +3\right )}{\left (-2 x^{2}+5 x +1\right )^{2}}d x\]

Input:

int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x)
 

Output:

int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x)
 

Fricas [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int { \frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{2}} \,d x } \] Input:

integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x, algorithm="fr 
icas")
 

Output:

integral((4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(4*x^4 - 20*x^3 + 21*x^2 + 
 10*x + 1), x)
 

Sympy [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int \frac {\left (- 6 x^{2} + 3 x + 7\right )^{q} \left (4 x^{2} + 2 x + 3\right )}{\left (2 x^{2} - 5 x - 1\right )^{2}}\, dx \] Input:

integrate((-6*x**2+3*x+7)**q*(4*x**2+2*x+3)/(-2*x**2+5*x+1)**2,x)
 

Output:

Integral((-6*x**2 + 3*x + 7)**q*(4*x**2 + 2*x + 3)/(2*x**2 - 5*x - 1)**2, 
x)
 

Maxima [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int { \frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{2}} \,d x } \] Input:

integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x, algorithm="ma 
xima")
 

Output:

integrate((4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(2*x^2 - 5*x - 1)^2, x)
 

Giac [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int { \frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{2}} \,d x } \] Input:

integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x, algorithm="gi 
ac")
 

Output:

integrate((4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(2*x^2 - 5*x - 1)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\int \frac {\left (4\,x^2+2\,x+3\right )\,{\left (-6\,x^2+3\,x+7\right )}^q}{{\left (-2\,x^2+5\,x+1\right )}^2} \,d x \] Input:

int(((2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q)/(5*x - 2*x^2 + 1)^2,x)
 

Output:

int(((2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q)/(5*x - 2*x^2 + 1)^2, x)
 

Reduce [F]

\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^2} \, dx=\text {too large to display} \] Input:

int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^2,x)
 

Output:

(66*( - 6*x**2 + 3*x + 7)**q*q*x + 19*( - 6*x**2 + 3*x + 7)**q*q - 42*( - 
6*x**2 + 3*x + 7)**q*x - 63492*int(( - 6*x**2 + 3*x + 7)**q/(528*q**2*x**6 
 - 2904*q**2*x**5 + 3476*q**2*x**4 + 3014*q**2*x**3 - 3762*q**2*x**2 - 160 
6*q**2*x - 154*q**2 - 600*q*x**6 + 3300*q*x**5 - 3950*q*x**4 - 3425*q*x**3 
 + 4275*q*x**2 + 1825*q*x + 175*q + 168*x**6 - 924*x**5 + 1106*x**4 + 959* 
x**3 - 1197*x**2 - 511*x - 49),x)*q**4*x**2 + 158730*int(( - 6*x**2 + 3*x 
+ 7)**q/(528*q**2*x**6 - 2904*q**2*x**5 + 3476*q**2*x**4 + 3014*q**2*x**3 
- 3762*q**2*x**2 - 1606*q**2*x - 154*q**2 - 600*q*x**6 + 3300*q*x**5 - 395 
0*q*x**4 - 3425*q*x**3 + 4275*q*x**2 + 1825*q*x + 175*q + 168*x**6 - 924*x 
**5 + 1106*x**4 + 959*x**3 - 1197*x**2 - 511*x - 49),x)*q**4*x + 31746*int 
(( - 6*x**2 + 3*x + 7)**q/(528*q**2*x**6 - 2904*q**2*x**5 + 3476*q**2*x**4 
 + 3014*q**2*x**3 - 3762*q**2*x**2 - 1606*q**2*x - 154*q**2 - 600*q*x**6 + 
 3300*q*x**5 - 3950*q*x**4 - 3425*q*x**3 + 4275*q*x**2 + 1825*q*x + 175*q 
+ 168*x**6 - 924*x**5 + 1106*x**4 + 959*x**3 - 1197*x**2 - 511*x - 49),x)* 
q**4 + 150382*int(( - 6*x**2 + 3*x + 7)**q/(528*q**2*x**6 - 2904*q**2*x**5 
 + 3476*q**2*x**4 + 3014*q**2*x**3 - 3762*q**2*x**2 - 1606*q**2*x - 154*q* 
*2 - 600*q*x**6 + 3300*q*x**5 - 3950*q*x**4 - 3425*q*x**3 + 4275*q*x**2 + 
1825*q*x + 175*q + 168*x**6 - 924*x**5 + 1106*x**4 + 959*x**3 - 1197*x**2 
- 511*x - 49),x)*q**3*x**2 - 375955*int(( - 6*x**2 + 3*x + 7)**q/(528*q**2 
*x**6 - 2904*q**2*x**5 + 3476*q**2*x**4 + 3014*q**2*x**3 - 3762*q**2*x*...